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DOE OSTI2022

High Order Wall-Modeled Large-Eddy Simulation on Mixed Unstructured Meshes

In the present study, an algebraic equilibrium wall model previously developed for hexahedral elements is extended to handle mixed meshes including prismatic, tetrahedral, and pyramidal elements in the context of a discontinuous high-order method. This extension is needed for complex geometries, for which high-order mixed elements (e.g., tetrahedral and pyramidal elements) are often necessary near solid walls to avoid meshing challenges. Various design decisions are discussed to achieve the best performance on massively parallel CPU/GPU architectures for a production-level high-order large-eddy simulation solver based on the flux reconstruction/correction procedure via reconstruction method, hpMusic. The extension to other elements is first evaluated using a benchmark channel flow problem at various Reynolds numbers. After that, flow over the NASA high-lift Common Research Model (CRM-HL) from the 4th AIAA High-lift Prediction Workshop is computed to further test the new implementation. Computational results at the third- and fourth-order accuracies are compared with experimental data.

Engineering↗

NASA NTRS1994

Analytical and Experimental Vibration Analysis of a Faulty Gear System

A comprehensive analytical procedure was developed for predicting faults in gear transmission systems under normal operating conditions. A gear tooth fault model is developed to simulate the effects of pitting and wear on the vibration signal under normal operating conditions. The model uses changes in the gear mesh stiffness to simulate the effects of gear tooth faults. The overall dynamics of the gear transmission system is evaluated by coupling the dynamics of each individual gear-rotor system through gear mesh forces generated between each gear-rotor system and the bearing forces generated between the rotor and the gearbox structure. The predicted results were compared with experimental results obtained from a spiral bevel gear fatigue test rig at NASA Lewis Research Center. The Wigner-Ville distribution (WVD) was used to give a comprehensive comparison of the predicted and experimental results. The WVD method applied to the experimental results were also compared to other fault detection techniques to verify the WVD's ability to detect the pitting damage, and to determine its relative performance. Overall results show good correlation between the experimental vibration data of the damaged test gear and the predicted vibration from the model with simulated gear tooth pitting damage. Results also verified that the WVD method can successfully detect and locate gear tooth wear and pitting damage.

Choy, F. K.↗

NASA NTRS1998

The Tera Multithreaded Architecture and Unstructured Meshes

The Tera Multithreaded Architecture (MTA) is a new parallel supercomputer currently being installed at San Diego Supercomputing Center (SDSC). This machine has an architecture quite different from contemporary parallel machines. The computational processor is a custom design and the machine uses hardware to support very fine grained multithreading. The main memory is shared, hardware randomized and flat. These features make the machine highly suited to the execution of unstructured mesh problems, which are difficult to parallelize on other architectures. We report the results of a study carried out during July-August 1998 to evaluate the execution of EUL3D, a code that solves the Euler equations on an unstructured mesh, on the 2 processor Tera MTA at SDSC. Our investigation shows that parallelization of an unstructured code is extremely easy on the Tera. We were able to get an existing parallel code (designed for a shared memory machine), running on the Tera by changing only the compiler directives. Furthermore, a serial version of this code was compiled to run in parallel on the Tera by judicious use of directives to invoke the "full/empty" tag bits of the machine to obtain synchronization. This version achieves 212 and 406 Mflop/s on one and two processors respectively, and requires no attention to partitioning or placement of data issues that would be of paramount importance in other parallel architectures.

Bokhari, Shahid H.↗

DOE OSTI2014

The simulation of 3D mass models in 2D digital mammography and breast tomosynthesis

Purpose: This work proposes a new method of building 3D breast mass models with different morphological shapes and describes the validation of the realism of their appearance after simulation into 2D digital mammograms and breast tomosynthesis images. Methods: Twenty-five contrast enhanced MRI breast lesions were collected and each mass was manually segmented in the three orthogonal views: sagittal, coronal, and transversal. The segmented models were combined, resampled to have isotropic voxel sizes, triangularly meshed, and scaled to different sizes. These masses were referred to as nonspiculated masses and were then used as nuclei onto which spicules were grown with an iterative branching algorithm forming a total of 30 spiculated masses. These 55 mass models were projected into 2D projection images to obtain mammograms after image processing and into tomographic sequences of projection images, which were then reconstructed to form 3D tomosynthesis datasets. The realism of the appearance of these mass models was assessed by five radiologists via receiver operating characteristic (ROC) analysis when compared to 54 real masses. All lesions were also given a breast imaging reporting and data system (BIRADS) score. The data sets of 2D mammography and tomosynthesis were read separately. The Kendall's coefficient of concordance was used for the interrater observer agreement assessment for the BIRADS scores per modality. Further paired analysis, using the Wilcoxon signed rank test, of the BIRADS assessment between 2D and tomosynthesis was separately performed for the real masses and for the simulated masses. Results: The area under the ROC curves, averaged over all observers, was 0.54 (95% confidence interval [0.50, 0.66]) for the 2D study, and 0.67 (95% confidence interval [0.55, 0.79]) for the tomosynthesis study. According to the BIRADS scores, the nonspiculated and the spiculated masses varied in their degrees of malignancy from normal (BIRADS 1) to highly suggestive for malignancy (BIRADS 5) indicating the required variety of shapes and margins of these models. The assessment of the BIRADS scores for all observers indicated good agreement based on Kendall's coefficient for both the 2D and the tomosynthesis evaluations. The paired analysis of the BIRADS scores between 2D and tomosynthesis for each observer revealed consistent behavior for the real and simulated masses. Conclusions: A database of 3D mass models, with variety of shapes and margins, was validated for the realism of their appearance for 2D digital mammography and for breast tomosynthesis. This database is suitable for use in future observer performance studies whether in virtual clinical trials or in patient images with simulated lesions.

60 APPLIED LIFE SCIENCES↗

DOE OSTI2023

SPH modeling of biomass granular flow: Theoretical implementation and experimental validation

The commercialization of biomass-derived energy is impeded by flowability challenges arising from the feeding and handling of granular biomass materials in full-scale biorefineries. To overcome these obstacles, a robust and accurate model to simulate the flow of granular biomass is indispensable. However, conventional mesh-based numerical codes are limited by inherent mesh distortion in simulating large deformation that commonly occurs in granular biomass handling. Here, in this study, we propose a graphics processing unit (GPU)-accelerated meshless Smoothed Particle Hydrodynamics (SPH) code to model the flow of granular biomass materials. A modified void ratio-based mass conversation, a hybrid particle-to-particle/surface frictional boundary treatment, and a hypoplastic constitutive model are implemented. Four numerical examples, an elastic block sliding on inclined planes, sand column collapse, Angle of Repose, and axial compression tests for pine chips, were simulated using the developed SPH code. The results demonstrate good agreement between numerical predictions and analytical and experimental data for all four examples, validating the SPH code and increasing confidence that it can be applied to simulate more complex granular biomass handling processes, such as hopper feeding or auger conveyance.

09 BIOMASS FUELS↗

DOE OSTI2017

Simulating the dust content of galaxies: successes and failures

We present full-volume cosmological simulations, using the moving-mesh code AREPO to study the coevolution of dust and galaxies. We extend the dust model in AREPO to include thermal sputtering of grains and investigate the evolution of the dust mass function, the cosmic distribution of dust beyond the interstellar medium and the dependence of dust-to-stellar mass ratio on galactic properties. The simulated dust mass function is well described by a Schechter fit and lies closest to observations at z = 0. The radial scaling of projected dust surface density out to distances of 10 Mpc around galaxies with magnitudes 17 < i < 21 is similar to that seen in Sloan Digital Sky Survey data, albeit with a lower normalization. At z = 0, the predicted dust density of Ωdust ≈ 1.3 × 10 -6 lies in the range of Ωdust values seen in low-redshift observations. We find that the dust-to-stellar mass ratio anticorrelates with stellar mass for galaxies living along the star formation main sequence. Moreover, we estimate the 850 μm number density functions for simulated galaxies and analyse the relation between dust-to-stellar flux and mass ratios at z = 0. At high redshift, our model fails to produce enough dust-rich galaxies, and this tension is not alleviated by adopting a top-heavy initial mass function. We do not capture a decline in Ωdust from z = 2 to 0, which suggests that dust production mechanisms more strongly dependent on star formation may help to produce the observed number of dusty galaxies near the peak of cosmic star formation.

79 ASTRONOMY AND ASTROPHYSICS↗

NASA NTRS1992

The Keck keyword layer

Each Keck instrument presents a consistent software view to the user interface programmer. The view consists of a small library of functions, which are identical for all instruments, and a large set of keywords, that vary from instrument to instrument. All knowledge of the underlying task structure is hidden from the application programmer by the keyword layer. Image capture software uses the same function library to collect data for the image header. Because the image capture software and the instrument control software are built on top of the same keyword layer, a given observation can be 'replayed' by extracting keyword-value pairs from the image header and passing them back to the control system. The keyword layer features non-blocking as well as blocking I/O. A non-blocking keyword write operation (such as setting a filter position) specifies a callback to be invoked when the operation is complete. A non-blocking keyword read operation specifies a callback to be invoked whenever the keyword changes state. The keyword-callback style meshes well with the widget-callback style commonly used in X window programs. The first keyword library was built for the two Keck optical instruments. More recently, keyword libraries have been developed for the infrared instruments and for telescope control. Although the underlying mechanisms used for inter-process communication by each of these systems vary widely (Lick MUSIC, Sun RPC, and direct socket I/O, respectively), a basic user interface has been written that can be used with any of these systems. Since the keyword libraries are bound to user interface programs dynamically at run time, only a single set of user interface executables is needed. For example, the same program, 'xshow', can be used to display continuously the telescope's position, the time left in an instrument's exposure, or both values simultaneously. Less generic tools that operate on specific keywords, for example an X display that controls optical instrument exposures, have also been written using the keyword layer.

Conrad, A. R.↗

NASA NTRS1999

Crack Growth Prediction Methodology for Multi-Site Damage: Layered Analysis and Growth During Plasticity

A finite element program has been developed to perform quasi-static, elastic-plastic crack growth simulations. The model provides a general framework for mixed-mode I/II elastic-plastic fracture analysis using small strain assumptions and plane stress, plane strain, and axisymmetric finite elements. Cracks are modeled explicitly in the mesh. As the cracks propagate, automatic remeshing algorithms delete the mesh local to the crack tip, extend the crack, and build a new mesh around the new tip. State variable mapping algorithms transfer stresses and displacements from the old mesh to the new mesh. The von Mises material model is implemented in the context of a non-linear Newton solution scheme. The fracture criterion is the critical crack tip opening displacement, and crack direction is predicted by the maximum tensile stress criterion at the crack tip. The implementation can accommodate multiple curving and interacting cracks. An additional fracture algorithm based on nodal release can be used to simulate fracture along a horizontal plane of symmetry. A core of plane strain elements can be used with the nodal release algorithm to simulate the triaxial state of stress near the crack tip. Verification and validation studies compare analysis results with experimental data and published three-dimensional analysis results. Fracture predictions using nodal release for compact tension, middle-crack tension, and multi-site damage test specimens produced accurate results for residual strength and link-up loads. Curving crack predictions using remeshing/mapping were compared with experimental data for an Arcan mixed-mode specimen. Loading angles from 0 degrees to 90 degrees were analyzed. The maximum tensile stress criterion was able to predict the crack direction and path for all loading angles in which the material failed in tension. Residual strength was also accurately predicted for these cases.

James, Mark Anthony↗

NASA NTRS2006

Methodology for Life Testing of Refractory Metal/Sodium Heat Pipes

The focus of this work was to establish an approach to generate carefully controlled data that can conclusively establish heat pipe operating life with material-fluid combinations capable of extended operation. To accomplish this goal acceleration is required to compress 10 years of operational life into 3 years of laboratory testing through a combination of increased temperature and mass fluence. Specific test series have been identi3ed, based on American Society for Testing and Materials (ASTM) specifications, to investigate long term corrosion rates. The refractory metal selected for demonstration purposes is a Molybdenum-44.5%Rhenium alloy formed by powder metallurgy. The heat pipe makes use of an annular crescent wick design formed by hot isostatic pressing of Molybdenum-Rhenium wire mesh. The heat pipes are filled using vacuum distillation and purity sampling is considered. Testing of these units is round-the-clock with 6-month destructive and non-destructive inspection intervals to identify the onset and level of corrosion. Non-contact techniques are employed for providing power to the evaporator (radio frequency induction heating at I to 5 kW per unit) and calorimetry at the condenser (static gas gap coupled water cooled calorimeter). The planned operating temperature range would extend from 1123 to 1323 K. Accomplishments prior to project cancellation included successful demonstration of the heat pipe wick fabrication technique, establishment of all engineering designs, baselined operational test requirements and procurement/assembly of supporting test hardware systems.

Martin, James J.↗

NASA NTRS2018

Computational Component Build-Up for the X-57 Distributed Electric Propulsion Aircraft

A computational study of the wing for the distributed electric propulsion X-57 Maxwell airplane configuration at cruise and takeoff/landing conditions was completed. Three unstructured-mesh, Navier-Stokes computational fluid dynamics methods, FUN3D, USM3D and Kestrel, were used to predict the performance buildup of components to the full X-57 configuration. The goal of the X-57 wing and distributed electric propulsion system design was to meet or exceed the required lift coefficient of 3.95 for a stall speed of 58 knots. The X-57 Maxwell airplane was designed with a small, high aspect ratio cruise wing that was designed for a high cruise lift coefficient of 0.75 at a cruise speed of 150 knots and altitude of 8,000 ft, with an angle of attack of approximately 0deg. The computational data indicates that the X-57 full aircraft drag would meet the cruise drag goal with a 25 count drag margin. The cruise configuration maximum lift coefficient is 2.07 and without including the stabilator is 1.86 at an angle of attack of 14 deg, predicted with the USM3D flow solver using the Spalart-Allmaras turbulence model. The maximum lift coefficient for the high-lift wing (with the 30deg flap deflection) without the stabilator contribution is 2.60 at an angle of attack of 13 deg. For high-lift blowing conditions with 13.7 hp/prop, the maximum lift coefficient excluding the stabilator is 4.426 at (alpha) = 13 deg. Therefore, the lift augmentation from the high-lift propellers is 1.7 and the total lift augmentation from the high-lift system (30 deg flap deflection and the high-lift blowing) is 2.38. The drag for the high-lift wing with 30 deg flap deflection is much higher than the cruise wing configuration, but the high-lift system is used only during a small portion of the flight envelope. The pitching moment is relatively constant for both blown and unblown conditions when the stabilator is excluded. Modeling the full geometry has indicated some adverse effects from the fuselage on the wing and stabilator. At high angles of attack, the solutions with the USM3D flow solver using the Spalart-Allmaras turbulence model indicates large flow separation on the wing upper surface between the two high-lift nacelles near the fuselage, and also a reduction in sectional lift on the stabilator in the first 50 percent of the stabilator semispan. However, the large flow separation near the fuselage is mostly eliminated in the solutions predicted with two codes, USM3D and Kestrel, using Hybrid Reynolds-averaged Navier Stokes/Large Eddy Simulation turbulence models.

Deere, Karen A.↗

DOE OSTI2023

Release on the Virtual Test Bed of an MSRE thermal hydraulics model

The nuclear industry is taking leaps in innovations with companies seeking a sustainable energy future through advanced nuclear reactors. The \gls{DOE}’s \gls{neams} program seeks to substantiate and bolster the deployment of advanced reactors through flexible multifidelity, multiphysics simulations of advanced nuclear reactors. Applications like SAM for one-dimensional systems thermal-hydraulics, and Pronghorn for multidimensional coarse mesh thermal-hydraulics, are geared to support innovations in industry by facilitating design, optimization, and licensing of advanced nuclear reactors. Coupling systems thermal-hydraulics and computational fluid dynamics codes can be difficult as the pressure coupling converges slowly; however, it is important to obtain the desired accuracy in each part of the primary loop. The authors of this model created an \gls{odc}~\cite{Mau23} approach to coupling SAM and Pronghorn. Leveraging this coupling technique, a \gls{msre} model was developed and released to the \gls{neams}/\gls{nric} \gls{vtb}. The \gls{msre} was chosen to be modeled because of the wealth of experimental data available and because of the strong physics coupling between the core and primary circuit \cite{doi:10.13182/NT8-2-118}. This document contextualizes the history of the \gls{msre}, describes the thermal hydraulics models used, and detail the implementation of multidimensional thermal-hydraulics and system codes based on the \gls{odc} method~\cite{Penn} for the \gls{msre} model. Finally, this document presents how other modelers could apply the SAM-Pronghorn \gls{odc} for other advanced reactor models. Current item is the set of slides for ANS Winter 23. The release of the model and the ANS summary have already been approved

22 GENERAL STUDIES OF NUCLEAR REACTORS↗

DOE OSTI2023

Release on the Virtual Test Bed of an MSRE thermal hydraulics model

22 GENERAL STUDIES OF NUCLEAR REACTORS↗

DOE OSTI2026

Scaling kinetic Monte-Carlo simulations of grain growth with combined convolutional and graph neural networks

Graph neural networks (GNN) have emerged as a promising machine learning method for microstructure simulations such as grain growth. However, accurate modeling of realistic grain boundary networks requires large simulation cells, which GNN has difficulty scaling up to. To alleviate the computational costs and memory footprint of GNN, we suggest a hybrid architecture combining a convolutional neural network (CNN) based bijective autoencoder to compress the spatial dimensions, and a GNN that evolves the microstructure in the latent space of reduced spatial sizes. Our results demonstrate that the new design significantly reduces computational costs with using fewer message passing layer (from 12 down to 3) compared with GNN alone. The reduction in computational cost becomes more pronounced as the spatial size increases, indicating strong computational scalability. For the largest mesh evaluated (160 3 ), our method reduces memory usage and runtime in inference by 117× and 115×, respectively, compared with GNN-only baseline. More importantly, it shows higher accuracy and stronger spatiotemporal capability than the GNN-only baseline, especially in long-term testing. Such combination of scalability and accuracy is essential for simulating realistic material microstructures over extended time scales. The improvements can be attributed to the bijective autoencoder’s ability to compress information losslessly from spatial domain into a high dimensional feature space, thereby producing more expressive latent features for the GNN to learn from, while also contributing its own spatiotemporal modeling capability. Training data are generated from stochastic grain growth simulations, providing realistic variability for learning robust microstructure evolution. Comprehensive system validation confirms that the model is accurate, robust, and scalable.

36 MATERIALS SCIENCE↗

DOE OSTI2017

Steady RANS methodology for calculating pressure drop in an in-line molten salt compact crossflow heat exchanger

We report the development of molten salt cooled reactors (MSR) and fluoride-salt cooled high temperature reactors (FHR) requires the use of advanced design tools for the primary heat exchanger design. Due to geometric and flow characteristics, compact (pitch to diameter ratios equal to or less than 1.25) heat exchangers with a crossflow flow arrangement can become desirable for these reactors. Unfortunately, the available experimental data is limited for compact tube bundles or banks in crossflow. Computational Fluid Dynamics can be used to alleviate the lack of experimental data in these tube banks. Previous computational efforts have been primarily focused on large S/D ratios (larger than 1.4) using unsteady Reynolds averaged Navier-Stokes and Large Eddy Simulation frameworks. These approaches are useful, but have large computational requirements that make comprehensive design studies impractical. A CFD study was conducted with steady RANS in an effort to provide a starting point for future design work. The study was performed for an in-line tube bank geometry with FLiBe (LiF-BeF2), a frequently selected molten salt, as the working fluid. Based on the estimated pressure drops, the pressure and velocity distributions in the domain, an appropriate meshing strategy was determined and presented. Periodic boundaries in the spanwise direction transverse flow were determined to be an appropriate boundary condition for reduced computational domains. The domain size was investigated and a minimum of 2-flow channels for a domain is recommended to ensure the behavior is accounted for. Finally, the standard low Re κ-ε (Lien) turbulence model was determined to be the most appropriate for steady RANS of this case at the time of writing.

42 ENGINEERING↗

NASA NTRS1992

Cartesian based grid generation/adaptive mesh refinement

Grid adaptation has recently received attention in the computational fluid dynamics (CFD) community as a means to capture the salient features of a flowfield by either moving grid points of a structured or by adding cells in an unstructured manner. An approach based on a background cartesian mesh is investigated from which the geometry is 'cut' out of the mesh. Once the mesh is obtained, a solution on this coarse grid is found, that indicates which cells need to be refined. This process of refining/solving continues until the flow is grid refined in terms of a user specified global parameter (such as drag coefficient etc.). The advantages of this approach are twofold: the generation of the base grid is independent of the topology of the bodies or surfaces around/through which the flow is to be computed, and the resulting grid (in uncut regions) is highly isotropic, so that the truncation error is low. The flow solver (which, along with the grid generation is still under development) uses a completely unstructured data base, and is a finite volume, upwinding scheme. Current and future work will address generating Navier-Stokes suitable grids by using locally aligned and normal face/cell refining. The attached plot shows a simple grid about two turbine blades.

Coirier, William J.↗

NASA NTRS2015

Tribological Evaluation of Candidate Gear Materials Operating Under Light Loads in Highly Humid Conditions

A series of pin-on-disk sliding wear tests were undertaken to identify candidate materials for a pair of lightly loaded timing gears operating under highly humid conditions. The target application involves water purification and thus precludes the use of oil, grease and potentially toxic solid lubricants. The baseline sliding pair is austenitic stainless steel operating against a carbon filled polyimide. The test load and sliding speed (4.9N, 2.7ms) were chosen to represent average contact conditions of the meshing gear teeth. In addition to the baseline materials, the hard superelastic NiTiNOL 60 (60NiTi) was slid against itself, against the baseline polyimide, and against 60NiTi onto which a commercially deposited dry film lubricant (DFL) was applied. The alternate materials were evaluated as potential replacements to achieve a longer wear life and improved dimensional stability for the timing gear application. An attempt was also made to provide solid lubrication to self-mated 60NiTi by rubbing the polyimide against the disk wear track outside the primary 60NiTi-60NiTi contact, a method named stick or transfer-film lubrication. The selected test conditions gave repeatable friction and wear data and smooth sliding surfaces for the baseline materials similar to those in the target application. Friction and wear for self-mated stainless steel were high and erratic. Self-mated 60NiTi gave acceptably low friction (0.2) and modest wear but the sliding surfaces were rough and potentially unsuitable for the gear application. Tests in which 60NiTi pins were slid against DFL coated 60NiTi and DFL coated stainless steel gave low friction and long wear life. The use of stick lubrication via a secondary polyimide pin provided effective transfer film lubrication to self-mated 60NiTi tribological specimens. Using this approach, friction levels were equal or lower than the baseline polyimide-stainless combination and wear was higher but within data scatter observed in these preliminary tests. Based upon these results, self-mated 60NiTi gear teeth utilizing solid lubrication, is a reasonable approach for the target application.

tribology↗

NASA NTRS2015

Tribological Evaluation of Candidate Gear Materials Operating Under Light Loads in Highly Humid Conditions

A series of pin-on-disk sliding wear tests were undertaken to identify candidate materials for a pair of lightly loaded timing gears operating under highly humid conditions. The target application involves water purification and thus precludes the use of oil, grease and potentially toxic solid lubricants. The baseline sliding pair is austenitic stainless steel operating against a carbon filled polyimide. The test load and sliding speed (4.9 N, 2.7 m/s) were chosen to represent average contact conditions of the meshing gear teeth. In addition to the baseline materials, the hard superelastic NiTiNOL 60 (60NiTi) was slid against itself, against the baseline polyimide, and against 60NiTi onto which a commercially deposited dry film lubricant (DFL) was applied. The alternate materials were evaluated as potential replacements to achieve a longer wear life and improved dimensional stability for the timing gear application. An attempt was also made to provide solid lubrication to self-mated 60NiTi by rubbing the polyimide against the disk wear track outside the primary 60NiTi-60NiTi contact, a method named stick or transfer-film lubrication. The selected test conditions gave repeatable friction and wear data and smooth sliding surfaces for the baseline materials similar to those in the target application. Friction and wear for self-mated stainless steel were high and erratic. Self-mated 60NiTi gave acceptably low friction (approx. 0.2) and modest wear but the sliding surfaces were rough and potentially unsuitable for the gear application. Tests in which 60NiTi pins were slid against DFL coated 60NiTi and DFL coated stainless steel gave low friction and long wear life. The use of stick lubrication via the secondary polyimide pin provided effective transfer film lubrication to self-mated 60NiTi tribological specimens. Using this approach, friction levels were equal or lower than the baseline polyimide-stainless combination and wear was higher but within data scatter observed in these preliminary tests. Based upon these results, self-mated 60NiTi gear teeth utilizing solid lubrication, is a reasonable approach for the target application.

tribology↗

NASA NTRS1988

Accurate solutions, parameter studies and comparisons for the Euler and potential flow equations

Parameter studies are conducted using the Euler and potential flow equation models for unsteady and steady flows in both two and three dimensions. The Euler code is an implicit, upwind, finite volume code which uses the Van Leer method of flux-vector-splitting which has been recently extended for use on dynamic meshes and maintain all the properties of the original splitting. The potential flow code is an implicit, finite difference method for solving the transonic small disturbance equations and incorporates both entropy and vorticity corrections into the solution procedures thereby extending its applicability into regimes where shock strength normally precludes its use. Parameter studies resulting in benchmark type calculations include the effects of spatial and temporal refinement, spatial order of accuracy, far field boundary conditions for steady flow, frequency of oscillation, and the use of subiterations at each time step to reduce linearization and factorization errors. Comparisons between Euler and potential flows results are made as well as with experimental data where available.

Anderson, W. Kyle↗